What is the dramatic irony in "The False Gems"?
Let's define the concept first: the word 'irony' comes from from the Ancient Greek εἰρωνεία eirōneía, meaning dissimulation or feigned ignorance; 'irony' as a literary device implies ' an incongruity, or contrast, between what the expectations of a situation are and what is really the case, with a third element, that defines that what is really the case is ironic because of the situation that led to it.'
Now let's look at the story: our guy, Monsieur Lantin, has a wonderful wife who takes care of him and the house marvelously; she just happens to enjoy [what he believes are] immitation jeweleries and going to the theatre. He's teasing her about it; he resents that he has to go to the theatre with her from time to time; he resents her enjoying her jewelery, but she remains wonderfully gracious and loving... This is starting to look suspect.
Then she dies, he finds himself unable to keep a budget without her, incurs debts and decides to sell here jeweleries, which he expects would be worthless (that's his expectation of the situation). Much to his surprise, those were genuine, extremely valuable gems (there you go- incongruity with his expectations).
Now, what we need to find is the third term: this is ironic because of the situation that led to it: which boils down to: how could his wife have afforded genuine gems?
The most likely implication is that the wonderfully virtuous wife wasn't really the saint she made herself out to be. Could it be that she had a lover? More than one? Did her lovers give her all the valuable jewelery?? What was she doing while she was alegedly at the theatre without him? If he hadn't insisted on not going to the theatre with her, would she have remained faithful? (Remember that she started bringing home all the pretty jewels after he insisted she goes to the theatre with a friend rather than with him). In any case, she's leading a double life and he had no idea; hence it is ironic.